Stacks of black + white tetrahedrons awaiting assembly at the IFF for community building workshop.

These elegant towers of stacked tetrahedrons are the modular units from which we’ll be constructing a large-scale-model in Jake’s building workshop this afternoon. Made from 12 inch bamboo sticks, each tetrahedron consists of six sticks arranged with either a clockwise or anti clockwise rotation at each vertex. The innate chirality or  handedness of the tetrahedra causes stacks of these units to twist  into either a clockwise-spiraling triple helix, or an anticlockwise-spiraling triple helix. Chirality also has fundamental consequences for Dotson’s building methodology. If one builds solely from single handed tetrahedrons (what we can call homo-chirality), the resulting large scale structures will naturally curve and ultimately form into balls. If we mix left-handed and right-handed tetrahedrons, flat structures can be achieved. The choice of homo- or hetro-chirality has global architectural consequences.

Jake with spine towers awaiting assembly.

Below are pictures from the workshop. We were delighted to host a visit from Charles Long’s “Workshop Art” class from the University of California, Riverside. Students proved to be both adept and inventive: some followed Jakes purely algorithmic vision, other branched out on their own and went non-linear.

Workshop participants assembling modules.

Workshop participants constructing left and rich handed tetrahedrons for assembly into a larger network.

Children typically subvert algorithmic instructions and have no fear about launching off on their own. 9 year old Jaquell Weinraub  was no exception; he happily ignored his elders and went about building a starship.

Jaquell Weinraub building a starship.

Jaquell’s starship.

Margaret Wertheim and Tracy diligently assembling left-handed tetrahedrons.

Sheet of interlinking tetrahedrons.

When linking tetrahedrons together, a critical issue is the chirality of the modules. Here, the black tetrahedrons have vertices whose ends spiral in an anti-clockwise fashion, while the white tetrahedrons have  vertices that spiral clockwise. By slotting together vertices of alternating chirality one can construct a large  flat sheet. Note that the basic lattice structure here is hexagonal – commensurate with a Euclidean surface. Like a flat sheet of paper, these Euclidean trusses can be rolled into a cylinder.

Rolling a sheet of tetrahedra into a cylinder.

–Margaret Wertheim, Institute For Figuring

Stellated pentagons make a great spinning toy.

Pentagonal_toy_medium_res

There are lots of ways to connect sticks. A critical issue is the number of sticks that meet at each vertex. The form seen here is derived from the icosahedron, a Platonic solid that has five edges connecting at every vertex giving them pentagonal symmetry. Here, two pentagons – one orange and one blue – are woven together at the mid-plane, then each color is connected by a cone of the same-colored struts which all meet at a vertex. The resulting form is composed of two “stellated pentagons” offset at a 36 degree angle from one another. At our workshop today, the IFF’s Christina Simons realized that by putting a stick through the center the form could be turned into a charming spinning toy. It’s shadow reveals elegant symmetries hidden within the structure.

–Margaret Wertheim, Institute For Figuring

Jake Dotson with stacks on multicolored tetrahedrons.

Jake is installed in his residency and has been building amazing structures out of multicolored tetrahedrons. The tetrahedron – the simplest of the Platonic solids – can be constructed in two mirror-image configurations, in effect making for a left-handed and right-handed version. The two variations can be easily configured by a simple rearrangement of the colors of the struts. One of Jake’s research projects is an exploration of what kind of structures can be built if you only use either left-handed or right-handed tetrahedra. One early discovery is that when placed in stacks of only one handedness, the tetrahedra form towers that naturally twist into a triple-helix. One tower twists clockwise, the other twists anti-clockwise.

The handedness of tetrahedrons turns out to be central to organic chemistry, which is based on carbon, an atom that forms naturally into tetrahedral molecules. In chemistry one talks about the “chirality” of a molecule, referring to the fact that there can be left-handed and right-handed forms. A chiral molecule will have two distinct forms – known as “enantiomers” – both of which have the same chemical formulae. One entaniomer will be arranged in a left-handed fashion; the other will be right handed. The handedness or chirality of the molecule can be critical to its functioning. A famous case is the drug thalidomide: One of its enantiomers cured morning sickness in pregnant women, the other created birth defects in the fetus. Understanding chiral molecules is a major issue in drug design and in the basic biochemistry of living things.

— Margaret Wertheim, Institute For Figuring

That’s the big news from my first week in residence at the IFF (Well, the first big news, anyway).  Maybe not necessarily surprising, but still pretty exciting nonetheless!

I’ve been working with twisted tetrahedra in 3-D arrays for some time now, but always in arrangements that alternate between left-handed and right-handed orientations.  I call those hetero- (different) -chiral (handed) arrays and I’ve been really, really into them.  They make pretty straight lines in twelve different directions, and when I fit them together they feel fulfilled.  (Seriously, I don’t even think I’m projecting here, as I can say clearly and distinctly, that I also feel fulfilled)  And, heterochiral arrays of tetrahedra make a space filler (infinite 3d tiling of space): tetrahedra and truncated tetrahedra. So, there’s that as well (which is basically infinity).  But, I was never really interested in working with homo- (same) -chiral (handed) tetrahedral arrays until last week when I started making shapes here in the IFF Space.  I guess I was saving it for something special.  It turns out that homochiral tetrahedra are absolutely amazing when they are joined!  You just have to give them the space and support to do their own thing, which turns out to be: curve!  How about that!

So, now I’ve got a partially stellated icosadodecahedron (20 triangles/tetrahedra and 12 pentagons; pictured on the left, in blue and yellow), as well as a partially stellated truncated pentakisdodecahedron (At least, I think that’s what it is . . . 60 triangles/tetrahedra, 12 pentagons, and 20 hexagons; on the right in black and white) But wait! There’s more! The partially stellated icosadodecahedral array is also a space filler, as the stellations are in fact tetrahedra on the faces of a regular icosadodecahedron.  Boom! Again with the infinity! I have a sneaking suspicion that the partially stellated truncated pentakisdodecahedron has space filling potential too, but we’ll just have to see about that.

I’m so looking forward to sharing all this, and more, at the space filling workshop we’re having this weekend! It’s such an exciting time to be with tetrahedra.

I’m so happy to be here.

-Jake Dotson

Welcome to the blog for Jake Dotson’s Science + Art Residency, entitled Liberation Geometry. During his time at the IFF, from October 10 – December 21, 2013, Jake will keep us up-to-date on what he’s designing, building, and theorizing. Check back frequently for new images of his daily experiments. Check our Events page for a schedule of workshops and talks in conjunction with Jake’s residency.